Question

Given the first two terms of an infinitely decreasing geometric progression $\sqrt{3},\frac{2}{\sqrt{3}+1}$.

Find the common ratio and the sum of the progression.

Answer 1

Given two terms are $\sqrt{3},\frac{2}{\sqrt{3}+1}$

Common ratio is $\frac{2}{3+\sqrt{3}}$

Sum of given infinite series is

$\frac{\sqrt{3}}{1-\frac{2}{3+\sqrt{3}}}=\frac{3(1+\sqrt{3})}{1+\sqrt{3}}=3$